Ninth Edition

Farid Golnaraghi • Benjamin C. Kuo

Laplace Transform Table

Laplace Transform F(s)

1

1 s

1

1 i' + Q!

1

(s + a)2

n\

(s + a)" + 1

1 (s + a)(s + p)

s (s + a)(s + p)

1 s(s + a)

1

5(5 + a) 2

1 s2 (s + a)

1

s2(s + a)2

s

(s + a)2

CO,,

s2 + col

s s2+co2

Time Function f(t)

Unit-impulse function 8{t)

Unit-step function Us{i)

Unit-ramp function t

f(n = positive integer)

„-«' e

fa—at te

t"e~°"(n = positive integer)

1 (r-M r~Pl\{n=Lt {S-cSC C ){a*J

1 fi-o

0

pe-P-ae-*)^®

a

i ( l - e~at - ate'0") or

L{ca-\ + e-m) or

1 a2 a \ ct/

(1 -at)e-°"

sin co„t

I W W f l l

Laplace Transform Table (cont.)

Laplace Transform F(s)

4 s{s2+co2)

COn *-**rt

(5 + a) (j2 + ^ 2 )

w2

"'B J 2 + 2t;cons + co2

•4 s{s2 + 2t;cons + co2)

2 SO),,

s2 + 2£co„s + co2

coj,{s + a)

s2 + 2$co„s + a>2

-I s2(s2 + 2^cons + co2)

Time Function /(7)

1 — cos cont

con J a2 + OJ2 sin(6>„/ + 9)

where 6 = tan - ' (con/a)

W" r-°" 1 * nnf/.t t A1 <* + «5 \/ff2 + co2

w h e r e # = t a n - (co„/a)

"*" „ - f a ) „ / o i n , . , / i v-2 » ( r s \ \ ———e s i n a ; „ v J C ' (G<1)

1 ' i ?-&"'<\«{„hl^f\ & t +a)

wherefl^cos"""1? ( f < l )

2 _u>» r " t o " ' nn fa ) \ / l f2 f fll

wherefl = cos~l£ ( ? < 1 )

/ 2 -5

• W " " f ^ " ^ * ^ " " * ^ ^ 1 - f 2 , + " ) V '

where* - tan"1 ^ 1 - ^ (*<1) a - £con

O j - 1 , .

t - ^ + r ? ' - * * sm(w„\ / l - <2 / + 0) «h tony/I-^ v 7

where# = c o s _ l ( 2 £ 2 - l ) ( f < l )

Automatic Control Systems

FARID GOLNARAGHI Simon Fraser University

BENJAMIN C. KUO University of Illinois at Urbana-Champaign

WILEY

JOHN WILEY & SONS, INC.

VP & Executive Publisher Don Fowley Associate Publisher Daniel Sayre Senior Production Editor Nicole Repasky Marketing Manager Christopher Ruel Senior Designer Kevin Murphy Production Management Services Elm Street Publishing Services Editorial Assistant Carolyn Weisman Media Editor Lauren Sapira Cover Photo Science Source/Photo Researchers

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MATLAB* and Simullnk* are trademarks of The MathWorks. Inc. and are used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this hook. This book'S use or discussion of MATIAB* software or related products does not constitute endorsement or sponsorship hy The MathWorks of a particular pedagogical approach or particular use of the MATLAB software.

ISBN-13 978-0470-04896-2

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

To my wife, Mitra, and to Sophia and Carmen, the joys of my life. —M. Farid Golnaraghi

Preface (Readme)

This is the ninth edition of the text but the first with Farid Golnaraghi as the lead author. For this edition, we increased the number of examples, added MATLAB'" ' toolboxes, and enhanced the MATLAB GUI software, ACSYS. We added more computer-aided tools for students and teachers. The prepublication manuscript was reviewed by many professors, and most of the relevant suggestions have been adopted. In this edition, Chapters 1 through 4 are organized to contain all background material, while Chapters 5 through 10 contain material directly related to the subject of control.

In this edition, the following materials have been moved into appendices on this book's Web site at www.wiley.com/college/golnaraghi.

Appendix A: Elementary Matrix Theory and Algebra

Appendix B: Difference Equations

Appendix C: Laplace Transform Table

Appendix D: z-Transform Table

Appendix E: Properties and Construction of the Root Loci

Appendix F: General Nyquist Criterion

Appendix G: ACSYS 2008: Description of the Software

Appendix H: Discrete-Data Control Systems

In addition, the Web site contains the MATLAB files for ACSYS, which are software tools for solving control-system problems, and PowerPoint files for the illustrations in the text.

The following paragraphs are aimed at three groups: professors who have adopted the book or who we hope will select it as their text; practicing engineers looking for answers to solve their day-to-day design problems; and, finally, students who are going to live with the book because it has been assigned for the control-systems course they are taking.

To the Professor: The material assembled in this book is an outgrowth of senior-level control-system courses taught by the authors at their universities throughout their teaching careers. The first eight editions have been adopted by hundreds of universities in the United States and around the world and have been translated into at least six languages. Practically all the design topics presented in the eighth edition have been retained.

This text contains not only conventional MATLAB toolboxes, where students can learn MATLAB and utilize their programming skills, but also a graphical MATLAB-based software, ACSYS. The ACSYS software added to this edition is very different from the software accompanying any other control book. Here, through extensive use of MATLAB GUI programming, we have created software that is easy to use. As a result, students will need to focus only on learning control problems, not programming! We also have added two new applications, SIMLab and Virtual Lab, through which students work on realistic problems and conduct speed and position control labs in a software environment. In SIMLab, students have access to the system parameters and can alter them (as in any simulation). In Virtual Lab, we have introduced a black-box approach in which the students

iv

1 MATLAB " is a registered trademark of The Math Works. Inc.

Preface v

have no access to the plant parameters and have to use some sort of system identification technique to find them. Through Virtual Lab we have essentially provided students with a realistic online lab with all the problems they would encounter in a real speed- or position-control lab—for example, amplifier saturation, noise, and nonlinearity. We welcome your ideas for the future editions of this book.

Finally, a sample section-by-section for a one-semester course is given in the Instructor's Manual, which is available from the publisher to qualified instructors. The Manual also contains detailed solutions to all the problems in the book.

To Practicing Engineers: This book was written with the readers in mind and is very suitable for self-study. Our objective was to treat subjects clearly and thoroughly. The book does not use the theorem-proof-Q.E.D. style and is without heavy mathematics. The authors have consulted extensively for wide sectors of the industry for many years and have participated in solving numerous control-systems problems, from aerospace systems to industrial controls, automotive controls, and control of computer peripherals. Although it is difficult to adopt all the details and realism of practical problems in a textbook at this level, some examples and problems reflect simplified versions of real-life systems.

To Students: You have had it now that you have signed up for this course and your professor has assigned this book! You had no say about the choice, though you can form and express your opinion on the book after reading it. Worse yet, one of the reasons that your professor made the selection is because he or she intends to make you work hard. But please don't misunderstand us: what we really mean is that, though this is an easy book to study (in our opinion), it is a no-nonsense book. It doesn't have cartoons or nice-looking photographs to amuse you. From here on, it is all business and hard work. You should have had the prerequisites on subjects found in a typical linear-systems course, such as how to solve linear ordinary differential equations, Laplace transform and applications, and time-response and frequency-domain analysis of linear systems. In this book you will not find too much new mathematics to which you have not been exposed before. What is interesting and challenging is that you are going to learn how to apply some of the mathematics that you have acquired during the past two or three years of study in college. In case you need to review some of the mathematical foundations, you can find them in the appendices on this book's Web site. The Web site also contains lots of other goodies, including the ACSYS software, which is GUI software that uses MATLAB-based programs for solving linear control systems problems. You will also find the Simulink"12-based SIMLab and Virtual Lab, which will help you to gain understanding of real-world control systems.

This book has numerous illustrative examples. Some of these are deliberately simple for the purpose of illustrating new ideas and subject matter. Some examples are more elaborate, in order to bring the practical world closer to you. Furthermore, the objective of this book is to present a complex subject in a clear and thorough way. One of the important learning strategies for you as a student is not to rely strictly on the textbook assigned. When studying a certain subject, go to the library and check out a few similar texts to see how other authors treat the same subject. You may gain new perspectives on the subject and discover that one author may treat the material with more care and thoroughness than the others. Do not be distracted by written-down coverage with oversimplified examples. The minute you step into the real world, you will face the design of control systems with nonlinearities and/or time-varying elements as well as orders that can boggle your mind. It

2 Simulink'1* is a registered trademark of The Math Works, Inc.

may be discouraging to tell you now that strictly linear and first-order systems do not exist in the real world.

Some advanced engineering students in college do not believe that the material they learn in the classroom is ever going to be applied directly in industry. Some of our students come back from field and interview trips totally surprised to find that the material they learned in courses on control systems is actually being used in industry today. They are surprised to find that this book is also a popular reference for practicing engineers. Unfortunately, these fact-finding, eye-opening, and self-motivating trips usually occur near the end of their college days, which is often too late for students to get motivated.

There are many learning aids available to you: the MATLAB-based ACSYS software will assist you in solving all kinds of control-systems problems. The SIMLab and Virtual Lab software can be used for simulation of virtual experimental systems. These are all found on the Web site. In addition, the Review Questions and Summaries at the end of each chapter should be useful to you. Also on the Web site, you will find the errata and other supplemental material.

We hope that you will enjoy this book. It will represent another major textbook acquisition (investment) in your college career. Our advice to you is not to sell it back to the bookstore at the end of the semester. If you do so but find out later in your professional career that you need to refer to a control systems book, you will have to buy it again at a higher price.

Special Acknowledgments: The authors wish to thank the reviewers for their invaluable comments and suggestions. The prepublication reviews have had a great impact on the revision project.

The authors thank Simon Fraser students and research associates Michael Ages, Johannes Minor, Linda Franak, Arash Jamalian, Jennifer Leone, Neda Parnian, Sean MacPherson, Amin Kamalzadeh, and Nathan (Wuyang) Zheng for their help. Farid Golnaraghi also wishes to thank Professor Benjamin Kuo for sharing the pleasure of writing this wonderful book, and for his teachings, patience, and support throughout this experience.

M. F. Golnaraghi, Vancouver, British Columbia, Canada

B. C. Kuo, Champaign, Illinois, U.S.A.

2009

^Dr. Earl Foster, Dr. Vahe Caliskan,

^

Contents

Preface iv

> CHAPTER 1 Introduction 1

1-1 Introduction 1 1-1-1 Basic Components of a Control

System 2 1-1-2 Examples of Control-System

Applications 2 1-1-3 Open-Loop Control Systems

(Nonfeedback Systems) 5 1-1-4 Closed-Loop Control Systems

(Feedback Control Systems) 7 1-2 What Is Feedback, and What Are Its Effects? 8

1-2-1 Effect of Feedback on Overall Gain 8 1-2-2 Effect of Feedback on Stability 9 1-2-3 Effect of Feedback on External

Disturbance or Noise 10 1-3 Types of Feedback Control Systems 11

1-3-1 Linear versus Nonlinear Control Systems 11

1-3-2 Time-Invariant versus Time-Varying Systems 12

1-4 Summary 14

> CHAPTER 2

Mathematical Foundation 16

2-1 Complex-Variable Concept 16 2-1-1 2-1-2 2-1-3 2-1-4 2-1-5

2-1-6 2-1-7 Frequency 2-2-1

2-2-2 2-2-3

2-2-4 2-2-5

2-2-6 2-2-7

Complex Numbers 16 Complex Variables 18 Functions of a Complex Variable 19 Analytic Function 20 Singularities and Poles of a Function 20 Zeros of a Function 20 Polar Representation 22

-Domain Plots 26 Computer-Aided Construction of the Frequency-Domain Plots 26 Polar Plots 27 Bode Plot (Corner Plot or Asymptotic-Plot) 32 Real Constant K 34 Poles and Zeros at the Origin, (j<o)±p 34

Simple Zero, 1 +jtoT 37 Simple Pole, 1/(1 +jcoT) 39

2-2-8 Quadratic Poles and Zeros 39 2-2-9 Pure Time Delay, e^7-1 42 2-2-10 Magnitude-Phase Plot 44 2-2-11 Gain- and Phase-Crossover Points 46 2-2-12 Minimum-Phase and Nonminimum-

Phase Functions 47 2-3 Introdxiction to Differential Equations 49

2-3-1 Linear Ordinary Differential Equations 49

2-3-2 Nonlinear Differential Equations 49 2-3-3 First-Order Differential

Equations: State Equations 50 2-3-4 Definition of State Variables 50 2-3-5 The Output Equation 51

2-4 Laplace Transform 52 2-4-1 Definition of the Laplace

Transform 52 2-4-2 Inverse Laplace Transformation 54 2-4-3 Important Theorems of the Laplace

Transform 54 2-5 Inverse Laplace Transform by

Partial-Fraction Expansion 57 2-5-1 Partial-Fraction Expansion 57

2-6 Application of the Laplace Transform to the Solution of Linear Ordinary Differential Equations 62 2-6-1 First-Order Prototype System 63 2-6-2 Second-Order Prototype

System 64 2-7 Impulse Response and Transfer Functions

of Linear Systems 67 2-7-1 Impulse Response 67 2-7-2 Transfer Function (Single-Input,

Single-Output Systems) 70 2-7-3 Proper Transfer Functions 71 2-7-4 Characteristic Equation 71 2-7-5 Transfer Function (Multivariable

Systems) 71 2-8 Stability of Linear Control Systems 72 2-9 Bounded-Input, Bounded-Output

(BIBO) Stability—Continuous-Data Systems 73

2-10 Relationship between Characteristic Equation Roots and Stability 74

2-11 Zero-Input and Asymptotic Stability of Continuous-Data Systems 74

2-12 Methods of Determining Stability 77 2-13 Routh-Hurwitz Criterion 78

vii

vi i i Contents

2-13-1 Routh's Tabulation 79 2-13-2 Special Cases when Routh's

Tabulation Terminates Prematurely 80

2-14 MATLAB Tools and Case Studies 84 2-14-1 Description and Use of Transfer

Function Tool 84 2-14-2 MATLAB Tools for Stability 85

2-15 Summary 90

Block Diagrams and Signal-Flow Graphs 104

3-1 Block Diagrams 104 3-1-1

3-1-2

3-1-3 3-1-4

3-1-5

Typical Elements of Block Diagrams in Control Systems 106 Relation between Mathematical Equations and Block Diagrams 109 Block Diagram Reduction 113 Block Diagram of Multi-Input Systems—Special Case: Systems with a Disturbance 115 Block Diagrams and Transfer Functions of Multivariable Systems 117

3-2 Signal-Flow Graphs (SFGs) 119 3-2-1 Basic Elements of an SFG 119 3-2-2 Summary of the Basic Properties of

SFG 120 3-2-3 Definitions of SFG Terms 120 3-2-4 SFG Algebra 123 3-2-5 SFG of a Feedback Control

System 124 3-2-6 Relation between Block Diagrams

and SFGs 124 3-2-7 Gain Formula for SFG 124 3-2-8 Application of the Gain Formula

between Output Nodes and Noninpul Nodes 127

3-2-9 Application of the Gain Formula to Block Diagrams 128

3-2-10 Simplified Gain Formula 129 3-3 MATLAB Tools and Case Studies 129 3-4 Summary 133

• C H A P T E R 4

Theoretical Foundation and Background Material: Modeling of Dynamic Systems 147

4-1 Introduction to Modeling of Mechanical Systems 148 4-1-1 Translational Motion 148 4-1-2 Rotational Motion 157 4-1-3 Conversion between Translational and

Rotational Motions 161 4-1-4 Gear Trains 162

4-1-5 Backlash and Dead Zone (Nonlinear Characteristics) 164

4-2 Introduction to Modeling of Simple Electrical Systems 165 4-2-1 Modeling of Passive Electrical

Elements 165 4-2-2 Modeling of Electrical Networks 165

4-3 Modeling of Active Electrical Elements: Operational Amplifiers 172 4-3-1 The Ideal Op-Amp 173 4-3-2 Sums and Differences 173 4-3-3 First-Order Op-Amp

Configurations 174 4-4 Introduction to Modeling of Thermal Systems 177

4-4-1 Elementary Heat Transfer Properties 177

4-5 Introduction to Modeling of Fluid Systems ISO 4-5-1 Elementary Fluid and Gas System

Properties 180 4-6 Sensors and Encoders in Control Systems 189

4-6-1 Potentiometer 189 4-6-2 Tachometers 194 4-6-3 Incremental Encoder 195

4-7 DC Motors in Control Systems 198 4-7-1 Basic Operational Principles of DC

Motors 199 4-7-2 Basic Classifications of PM DC

Motors 199 4-7-3 Mathematical Modeling of PM DC

Motors 201 4-8 Systems with Transportation Lags

(Time Delays) 205 4-8-1 Approximation of the Time-Delay

Function bv Rational Functions 206

4-9 Linearization of Nonlinear Systems 206 4-9-1 Linearization Using Taylor Series:

Classical Representation 207 4-9-2 Linearization Using the State Space

Approach 207 4-10 Analogies 213 4-11 Case Studies 216 4-12 MATLAB Tools 222 4-13 Summary 223

Time-Domain Analysis of Control Systems 253

5-1 Time Response of Continuous-Data Systems: Introduction 253

5-2 Tvpical Test Signals for the Time Response of Control Systems 254

5-3 The Unit-Step Response and Time-Domain Specifications 256

5-4 Steady-State Error 258

Contents ix

5-4-1 Steady-State Error of Linear Continuous-Data Control Systems 258

5-4-2 Steady-State Error Caused by Nonlinear System Elements 272

5 Time Response of a Prototype First-Order System 274

6 Transient Response of a Prototype Seeond-Order System 275 5-6-1 Damping Ratio and Damping

Factor 277 5-6-2 Natural Undamped Frequency 278 5-6-3 Maximum Overshoot 280 5-6-4 Delay Time and Rise Time 283 5-6-5 Settling Time 285

7 Speed and Position Control of a DC Motor 289 5-7-1 Speed Response and the Effects of

Inductance and Disturbance-Open Loop Response 289

5-7-2 Speed Control of DC Motors: Closed-Loop Response 291

5-7-3 Position Control 292 8 Time-Domain Analysis of a Position-Control

System 293 5-8-1 Unit-Step Transient Response 294 5-8-2 The Steady-State Response 298 5-8-3 Time Response to a Unit-Ramp

Input 298 5-8-4 Time Response of a Third-Order

System 300 -9 Basic Control Systems and Effects of

Adding Poles and Zeros to Transfer Functions 304 5-9-1 Addition of a Pole to the

Forward-Path Transfer Function: Unit)'-Feedback Systems 305

5-9-2 Addition of a Pole to the Closed-Loop Transfer Function 307

5-9-3 Addition of a Zero to the Closed-Loop Transfer Function 308

5-9-4 Addition of a Zero to the Forward-Path Transfer Function: Unity-Feedback Systems 309

10 Dominant Poles and Zeros of Transfer Functions 311 5-10-1 Summary of Effects of Poles and

Zeros 313 5-10-2 The Relative Damping Ratio 313 5-10-3 The Proper Way of Neglecting the

Insignificant Poles with Consideration of the Steady-State Response 313

11 Basic Control Systems Utilizing Addition of Poles and Zeros 314

-12 MATLAB Tools 319 -13 Summary 320

The Control Lab 337

6-1 Introduction 337 6-2 Description of the Virtual Experimental

System 338 6-2-1 Motor 339 6-2-2 Position Sensor or Speed Sensor 339 6-2-3 Power Amplifier 340 6-2-4 Interface 340

6-3 Description of SIMLab and Virtual Lab Software 340

6-4 Simulation and Virtual Experiments 345 6-4-1 Open-Loop Speed 345 6-4-2 Open-Loop Sine Input 347 6-4-3 Speed Control .350 6-4-4 Position Control 352

6-5 Design Project 1—Robotic Arm 354 6-6 Design Project 2—Quarter-Car Model 357

6-6-1 Introduction to the Quarter-Car Model 357

6-6-2 Closed-Loop Acceleration Control 359

6-6-3 Description of Quarter Car Modeling Tool 360

6-6-4 Passive Suspension 364 6-6-5 Closed-Loop Relative Position

Control 365 6-6-6 Closed-Loop Acceleration

Control 366 6-7 Summary 367

Root Locus Analysis 372

7-1 Introduction 372 7-2 Basic Properties of the Root

Loci (RL) 373 7-3 Properties of the Root Loci 377

7-3-1 K = () and K = ±oo Points 377 7-3-2 Number of Branches on the Root

Loci 378 7-3-3 Symmetry of the RL 378 7-3-4 Angles of Asymptotes of the RL:

Behavior of the RL at |.y| = oo 378 7-3-5 Intersect of the Asymptotes

(Centroid) 379 7-3-6 Root Loci on the Real Axis 380 7-3-7 Angles of Departure and Angles of

Arrival of the RL 380 7-3-8 Intersection of the RL with the

Imaginary Axis 380 7-3-9 Breakaway Points (Saddle Points)

on the RL 380 7-3-10 The Root Sensitivity 382

x E> Contents

7-4 Design Aspects of the Root Loci 385 7-4-1 Effects of Adding Poles and Zeros

to G(s) H{s) 385 7-5 Root Contours (RC): Multiple-Parameter

Variation 393 7-6 MATLAB Tools and Case Studies 400 7-7 Summary 400

6- C H A P T E R 8

Frequency-Domain Analysis 409

8-1 Introduction 409 8-1-1 Frequency Response of

Closed-Loop Systems 410 8-1-2 Frequency-Domain Specifications 412

8-2 Mr, u>r, and Bandwidth of the Prototype Second-Order System 413 8-2-1 Resonant Peak and Resonant

Frequency 413 8-2-2 Bandwidth 416

8-3 Effects of Adding a Zero to the Forward-Path Transfer Function 418

8-4 Effects of Adding a Pole to the Forward-Path Transfer Function 424

8-5 Nyquist Stability Criterion: Fundamentals 426 8-5-1 Stability Problem 427 8-5-2 Definition of Encircled and

Enclosed 428 8-5-3 Number of Encirclements and

Enclosures 429 8-5-4 Principles of the Argument 429 8-5-5 Nyquist Path 433 8-5-6 Nyquist Criterion and the L(s) or

the G(s)H(s) Plot 434 8-6 Nyquist Criterion for Systems with

Minimum-Phase Transfer Functions 435 8-6-1 Application of the Nyquist Criterion

to Minimum-Phase Tranfer Functions That Are Not Strictly Proper 436

8-7 Relation between the Root Loci and the Nyquist Plot 437

8-8 Illustrative Examples: Nyquist Criterion for Minimum-Phase Transfer Functions 440

8-9 Effects of Adding Poles and Zeros to Lis) on the Shape of the Nyquist Plot 444

8-10 Relative Stability: Cain Margin and Phase Margin 449 8-10-1 Gain Margin (GM) 451 8-10-2 Phase Margin (PM) 453

8-11 Stability Analysis with the Bode Plot 455 8-11-1 Bode Plots of Systems with Pure

Time Delays 458

8-12 Relative Stability Related to the Slope of the Magnitude Curve of the Bode Plot 459 8-12-1 Conditionally Stable System 459

8-13 Stability Analysis with the Magnitude-Phase Plot 462

8-14 Constant-M Loci in the Magnitude-Phase Plane: The Nichols Chart 463

8-15 Nichols Chart Applied to Nonunity-Feedback Systems 469

8-16 Sensitivity Studies in the Frequency Domain 470 8-17 MATLAB Tools and Case Studies 472 8-18 Summary 472

Design of Control Systems 487

Introduction 487 9-1-1 9-1-2 9-1-3 Design 9-2-1

9-2-2

9-2-3 Design 9-3-1

9-3-2

Design Specifications 487 Controller Configurations 489 Fundamental Principles of Design 491

with the PD Controller 492 Time-Domain Interpretation of PD Control 494 Frequency-Domain Interpretation of PD Control 496 Summary of Effects of PD Control 497

with the PI Controller 511 Time-Domain Interpretation and Design of PI Control 513 Frequency-Domain Interpretation and Design of PI Control 514

9-4 Design with the PID Controller 528 9-5 Design with Phase-Lead Controller 532

9-5-1 Time-Domain Interpretation and Design of Phase-Lead Control 534

9-5-2 Frequency-Domain Interpretation and Design of Phase-Lead Control 535

9-5-3 Effects of Phase-Lead Compensation 554

9-5-4 Limitations of Single-Stage Phase-Lead Control 555

9-5-5 Multistage Phase-Lead Controller 555 9-5-6 Sensitivity Considerations 559

9-6 Design with Phase-Lag Controller 561 9-6-1 Time-Domain Interpretation and

Design of Phase-Lag Control 561 9-6-2 Frequency-Domain Interpretation

and Design of Phase-Lag Control 563 9-6-3 Effects and Limitations of Phase-Lag

Control 574 9-7 Design with Lead-Lag Controller 574 9-8 Pole-Zero-Cancellation Design: Notch Filter 576

9-8-J Second-Order Active Filter 579 9-8-2 Frequency-Domain Interpretation and

Design 580

Contents • x i

9-9 9-10 9-11

9-12

9-13

9-14 9-15 9-16

Forward and Feedforward Controllers 588 Design of Robust Control Systems 590 Minor-Loop Feedback Control 601 9-11-1 Rate-Feedback or

Tachometer-Feedback Control 601 9-11-2 Minor-Loop Feedback Control with

Active Filter 603 A Hydraulic Control System 605 9-12-1 Modeling Linear Actuator 605 9-12-2 Four-Way Electro-Hydraulic

Valve 606 9-12-3 Modeling the Hydraulic System 612 9-12-4 Applications 613 Controller Design 617 9-13-1 P Control 617 9-13-2 PD Control 621 9-13-3 PI Control 626 9-13-4 PID Control 628 MATLAB Tools and Case Studies 631 Plotting Tutorial 647 Summary 649

State Variable Analysis 673

10-1 Introduction 673 10-2 Block Diagrams, Transfer Functions, and State

Diagrams 673 10-2-1 Transfer Functions (Multivariable

Systems) 673 10-2-2 Block Diagrams and Transfer Functions

of Multivariable Systems 674 10-2-3 State Diagram 676 10-2-4 From Differential Equations to State

Diagrams 678 10-2-5 From State Diagrams to Transfer

Function 679 10-2-6 From State Diagrams to State and

Output Equations 680 10-3 Vector-Matrix Representation of State

Equations 682 10-4 State-Transition Matrix 684

10-4-1 Significance of the State-Transition Matrix 685

10-4-2 Properties of the State-Transition Matrix 685

10-5 State-Transition Equation 687 10-5-1 State-Transition Equation Determined

from the State Diagram 689 10-6 Relationship between State Equations and

High-Order Differential Equations 691 10-7 Relationship between State Equations and

Transfer Functions 693 10-8 Characteristic Equations, Eigenvalues,

and Eigenvectors 695

10-8-1 Characteristic Equation from a Differential Equation 695

10-8-2 Characteristic Equation from a Transfer Function 696

10-8-3 Characteristic Equation from State Equations 696

10-8-4 Eigenvalues 697 10-8-5 Eigenvectors 697 10-8-6 Generalized Eigenvectors 698

10-9 Similarity Transformation 699 10-9-1 Invariance Properties of the Similarity

Transformations 700 10-9-2 Controllability Canonical Form (CCF)

701 10-9-3 Observability Canonical Form (OCF) 703 10-9-4 Diagonal Canonical Form (DCF) 704 10-9-5 Jordan Canonical Form (JCF) 706 Decompositions of Transfer Functions 707 10-10-1 Direct Decomposition 707 10-10-2 Cascade Decomposition 712 10-10-3 Parallel Decomposition 713 Controllability of Control Systems 714 10-11-1 General Concept of Controllability'

716 10-11-2 Definition of State Controllability 10-11-3 Alternate Tests on Controllability Observability of Linear Systems 719 10-12-1 Definition of Observability 719 10-12-2 Alternate Tests on Observability Relationship among Controllability, Observability, and Transfer Functions 721 Invariant Theorems on Controllability and Observability 723 Case Study: Magnetic-Ball Suspension System 725 State-Feedback Control 728 Pole-Placement Design Through State Feedback 730 State Feedback with Integral Control 735 MATLAB Tools and Case Studies 741

10-10

10-11

10-12

10-13

10-14

10-15

10-16 10-17

10-18 10-19

716 717

720

10-19-1

10-19-2

10-20 Summary

Description and Use of the State-Space Analysis Tool 741 Description and Use of tfsym for State-Space Applications 748 751

" . ' • . : • - - • " :

Appendices can be found on this book's companion Web site: www.wiley.com/college/golnaraghi.

<-:?•••: 'T. : ... . . . r .

Elementary Matrix Theory and Algebra A-1

A-l Elementary Matrix Theory A-1 A-l-1 Definition of a Matrix A-2

xi i Contents

A-2 Matrix Algebra A-5 A-2-1 Equality of Matrices A-5 A-2-2 Addition and Subtraction of

Matrices A-6 A-2-3 Associative Law of Matrix (Addition and

Subtraction) A-6 A-2-4 Commutative Law of Matrix (Addition

and Subtraction) A-6 A-2-5 Matrix Multiplication A-6 A-2-6 Rules of Matrix Multiplication A-7 A-2-7 Multiplication by a Scalar k A-8 A-2-8 Inverse of a Matrix (Matrix Division) A-8 A-2-9 Rank of a Matrix A-8

A-3 Computer-Aided Solutions of Matrices A-9

- . - , " - - : • = « * • : • • . : r .

Difference Equations B-1

B-l Difference Equations B-1

Laplace Transform Table C-1

• • . . - ' > ; . 5 s : n • ; : : . :

z-Transform Table D-1

Properties and Construction of the Root Loci E-1

E-l K = 0 and K = ±00 Points E-1 E-2 Number of Branches on the Root Loci E-2 E-3 Symmetry' of the Root Loci E-2 E-4 Angles of Asymptotes of the Root Loci and

Behavior of the Root Loci at \s\ = 00 E-4 E-5 Intersect of the Asymptotes (Centroid) E-5 E-6 Root Loci on the Real Axis E-8 E-7 Angles of Departure and Angles of Arrival of the

Root Loci E-9 E-8 Intersection of the Root Loci with the

Imaginary Axis E- l l E-9 Breakaway Points E- l l

E-9-1 (Saddle Points) on the Root Loci E- l l E-9-2 The Angle of Arrival and Departure of

Root Loci at the Breakaway Point E-12 E-10 Calculation of K on the Root Loci E-16

General Nyquist Criterion F-l

F-l Formulation of Nyquist Criterion F-l F-l-1 System with Minimum-Phase Loop

Transfer Functions F-4 F-l-2 Systems with Improper Loop Transfer

Functions F-4 F-2 Illustrative Examples—General Nyquist Criterion

Minimum and Nonminimum Transfer Functions F-4 F-3 Stability Analysis of Multiloop Systems F-13

ACSYS 2008: Description of the Software G-1

G-l Installation of ACSYS G-1 G-2 Description of the Software G-1

G-2-1 tfsym G-2 G-2-2 Statetool G-3 G-2-3 Controls G-3 G-2-4 SIMLab and Virtual Lab G-4

G-3 Final Comments G-4

Discrete-Data Control Systems H-1

H-1 Introduction H-1 H-2 The z-Transform H-1

H-2-1 Definition of the z-Transform H-1 H-2-2 Relationship between the Laplace

Transform and the z-Transform H-2 H-2-3 Some Important Theorems of the

z-Transform H-3 H-2-4 Inverse z-Transform H-5 H-2-5 Computer Solution of the Partial-

Fraction Expansion of Y(z)/z H-7 H-2-6 Application of the z-Transform to the

Solution of Linear Difference Equations H-7

H-3 Transfer Functions of Discrete-Data Systems H-8 H-3-1 Transfer Functions of Discrete-Data

Systems with Cascade Elements H-12 H-3-2 Transfer Function of the Zero-Order-

Hold H-13 H-3-3 Transfer Functions of Closed-Loop

Discrete-Data Systems H-14 H-4 State Equations of Linear Discrete-Data

Systems H-16 H-4-1 Discrete State Equations H-16 H-4-2 Solutions of the Discrete State

Equations: Discrete State-Transition Equations H-18

H-4-3 z-Transform Solution of Discrete State Equations H-19

H-4-4 Transfer-Function Matrix and the Characteristic Equation H-20

H-4-5 State Diagrams of Discrete-Data Systems IT-22

H-4-6 State Diagrams for Sampled-Data Systems H-23

H-5 Stability of Discrete-Data Systems H-26 H-5-1 BIBO Stability H-26 H-5-2 Zero-Input Stability H-26 11-5-3 Stability Tests of Discrete-Data

Systems H-27 H-6 Time-Domain Properties of Discrete-Data

Contents xi i i

Systems H-31 H-6-1 Time Response of Discrete-Data

Control Systems H-31 H-6-2 Mapping between y-Plane and s-Plane

Trajectories H-34 H-6-3 Relation between Characteristic-

Equation Roots and Transient Response H-38

H-7 Steady-State Error Analysis of Discrete-Data Control Systems H-41

II-S Root Loci of Discrete-Data Systems H-45 H-9 Frequency-Domain Analysis of Discrete-Data

Control Systems H-49 H-9-1 Rode Plot with the

^'-Transformation H-50 H-10 Design of Discrete-Data Control Systems H-51

H-10-1 Introduction H-51

H-10-2 Digital Implementation of Analog Controllers H-52

H-10-3 Digital Implementation of the PID Controller H-54

H-10-4 Digital Implementation of Lead and Lag Controllers II-57

H-l l Digital Controllers 11-58 II - l l - l Physical Realizability of Digital

Controllers H-5S H-12 Design of Discrete-Data Control Systems in

the Frequency Domain and the z-Plane H-61 H-12-1 Phase-Lead and Phase-Lag Controllers

in the ic-Domain 11-61 H-13 Design of Discrete-Data Control Systems

with Deadbeat Response II-6S H-14 Pole-Placement Design with State

Feedback H-70

I CHAPTER

Introduction

• 1-1 INTRODUCTION

The main objectives of this chapter are:

1. To define a control system.

2. To explain why control systems are important.

3. To introduce the basic components of a control system.

4. To give some examples of control-system applications.

5. To explain why feedback is incorporated into most control systems.

6. To introduce types of control systems.

One of the most commonly asked questions by a novice on a control system is: What is a control system? To answer the question, we can say that in our daily lives there are numerous "objectives" that need to be accomplished. For instance, in the domestic domain, we need to regulate the temperature and humidity of homes and buildings for comfortable living. For transportation, we need to control the automobile and airplane to go from one point to another accurately and safely. Industrially, manufacturing processes contain numerous objectives for products that will satisfy the precision and cost-effectiveness requirements. A human being is capable of performing a wide range of tasks, including decision making. Some of these tasks, such as picking up objects and walking from one point to another, are commonly carried out in a routine fashion. Under certain conditions, some of these tasks are to be performed in the best possible way. For instance, an athlete running a 100-yard dash has the objective of running that distance in the shortest possible time. A marathon runner, on the other hand, not only must run the distance as quickly as possible, but, in doing so, he or she must control the consumption of energy and devise the best strategy for the race. The means of achieving these "objectives" usually involve the use of control systems that implement certain control strategies.

In recent years, control systems have assumed an increasingly important role in the development and advancement of modern civilization and technology. Practically every aspect of our day-to-day activities is affected by some type of control system. Control systems are found in abundance in all sectors of industry, such as quality control of manufactured products, automatic assembly lines, machine-tool control, space technology and weapon systems, computer control, transportation systems, power systems, robotics, Micro-Electro-Mechanical Systems (MEMS), nanotechnology, and many others. Even the control of inventory and social and economic systems may be approached from the theory of automatic control.

• Control systems are in abundance in modern civilization.

1

Chapter 1. Introduction

Objectives CONTROL SYSTEM

Results Figure 1-1 Basic components of a control system.

1 Basic Components of a Control System

The basic ingredients of a control system can be described by:

1. Objectives of control.

2. Control-system components.

3. Results or outputs.

The basic relationship among these three components is illustrated in Fig. 1 -1. In more technical terms, the objectives can be identified with inputs, or actuating signals, u, and the results are also called outputs, or controlled variables, y. In general, the objective of the control system is to control the outputs in some prescribed manner by the inputs through the elements of the control system.

2 Examples of Control-System Applications

Intelligent Systems Applications of control systems have significantly increased through the development of new materials, which provide unique opportunities for highly efficient actuation and sensing, thereby reducing energy losses and environmental impacts. State-of-the-art actuators and sensors may be implemented in virtually any system, including biological propulsion; locomotion; robotics; material handling; biomedical, surgical, and endoscopic; aeronautics; marine; and the defense and space industries. Potential applications of control of these systems may benefit the following areas:

• Machine tools. Improve precision and increase productivity by controlling chatter.

• Flexible robotics. Enable faster motion with greater accuracy.

• Photolithography. Enable the manufacture of smaller microelectronic circuits by controlling vibration in the photolithography circuit-printing process.

• Biomechanical and biomedical. Artificial muscles, drug delivery systems, and other assistive technologies.

• Process control. For example, on/off shape control of solar reflectors or aero­dynamic surfaces.

Control in Virtual Prototyping and Hardware in the Loop The concept of virtual prototyping has become a widely used phenomenon in the automotive, aerospace, defense, and space industries. In all these areas, pressure to cut costs has forced manufacturers to design and test an entire system in a computer environment before a physical prototype is made. Design tools such as MATLAB and Simulink enable companies to design and test controllers for different components (e.g., suspension, ABS, steering, engines, flight control mechanisms, landing gear, and special­ized devices) within the system and examine the behavior of the control system on the virtual prototype in real time. This allows the designers to change or adjust controller parameters online before the actual hardware is developed. Hardware in the loop terminology is a new approach of testing individual components by attaching them to the virtual and controller prototypes. Here the physical controller hardware is interfaced with the computer and replaces its mathematical model within the computer!

1-1 Introduction 3

Smart Transportation Systems The automobile and its evolution in the last two centuries is arguably the most transform­ative invention of man. Over years innovations have made cars faster, stronger, and aesthetically appealing. We have grown to desire cars that are "intelligent" and provide maximum levels of comfort, safety, and fuel efficiency. Examples of intelligent systems in cars include climate control, cruise control, anti-lock brake systems (ABSs), active suspensions that reduce vehicle vibration over rough terrain or improve stability, air springs that self-level the vehicle in high-G turns (in addition to providing a better ride), integrated vehicle dynamics that provide yaw control when the vehicle is either over- or understeering (by selectively activating the brakes to regain vehicle control), traction control systems to prevent spinning of wheels during acceleration, and active sway bars to provide "controlled" rolling of the vehicle. The following are a few examples.

Drive-by-wire and Driver Assist Systems The new generations of intelligent vehicles will be able to understand the driving environment, know their whereabouts, monitor their health, understand the road signs, and monitor driver performance, even overriding drivers to avoid catastrophic accidents. These tasks require significant overhaul of current designs. Drive-by-wire technology replaces the traditional mechanical and hydraulic systems with electronics and control systems, using electromechanical actuators and human-machine interfaces such as pedal and steering feel emulators—otherwise known as haptic systems. Hence, the traditional components—such as the steering column, intermediate shafts, pumps, hoses, fluids, belts, coolers, brake boosters, and master cylinders—are eliminated from the vehicle. Haptic interfaces that can offer adequate transparency to the driver while maintaining safety and stability of the system. Removing the bulky mechanical steering wheel column and the rest of the steering system has clear advantages in terms of mass reduction and safety in modern vehicles, along with improved ergonomics as a result of creating more driver space. Replacing the steering wheel with a haptic device that the driver controls through the sense of touch would be useful in this regard. The haptic device would produce the same sense to the driver as the mechanical steering wheel but with improvements in cost, safety, and fuel consumption as a result of eliminating the bulky mechanical system.

Driver assist systems help drivers to avoid or mitigate an accident by sensing the nature and significance of the danger. Depending on the significance and timing of the threat, these on-board safety systems will initially alert the driver as early as possible to an impending danger. Then, they will actively assist or, ultimately, intervene in order to avert the accident or mitigate its consequences. Provisions for automatic over-ride features when the driver loses control due to fatigue or lack of attention will be an important part of the system. In such systems, the so-called advanced vehicle control system monitors the longitudinal and lateral control, and by interacting with a central management unit, it will be ready to take control of the vehicle whenever the need arises. The system can be readily integrated with sensor networks that monitor every aspect of the conditions on the road and are prepared to take appropriate action in a safe manner.

Integration and Utilization of Advanced Hybrid Powertrains Hybrid technologies offer improved fuel consumption while enhancing driving experience. Utilizing new energy storage and conversion technologies and integrating them with powertrains would be prime objectives of this research activity. Such technologies must be compatible with current platforms and must enhance, rather than compromise, vehicle function. Sample applica­tions would include developing plug-in hybrid technology, which would enhance the vehicle cruising distance based on using battery power alone, and utilizing sustainable

energy resources, such as solar and wind power, to charge the batteries. The smart plug-in vehicle can be a part of an integrated smart home and grid energy system of the future, which would utilize smart energy metering devices for optimal use of grid energy by avoiding peak energy consumption hours.

High Performance Real-time Control, Health Monitoring, and Diagnosis Modern vehicles utilize an increasing number of sensors, actuators, and networked embedded computers. The need for high performance computing would increase with the introduction of such revolutionary features as drive-by-wire systems into modern vehicles. The tremendous computational burden of processing sensory data into appropriate control and monitoring signals and diagnostic information creates challenges in the design of embedded computing technology. Towards this end, a related challenge is to incorporate sophisticated computational techniques that control, monitor, and diagnose complex automotive systems while meeting requirements such as low power consumption and cost effectiveness.

The following represent more traditional applications of control that have become part of our daily lives.

Steering Control of an Automobile As a simple example of the control system, as shown in Fig. 1-1, consider the steering control of an automobile. The direction of the two front wheels can be regarded as the controlled variable, or the output, y; the direction of the steering wheel is the actuating signal, or the input, u. The control system, or process in this case, is composed of the steering mechanism and the dynamics of the entire automobile. However, if the objective is to control the speed of the automobile, then the amount of pressure exerted on the accelerator is the actuating signal, and the vehicle speed is the controlled variable. As a whole, we can regard the simplified automobile control system as one with two inputs (steering and accelerator) and two outputs (heading and speed). In this case, the two controls and two outputs are independent of each other, but there are systems for which the controls are coupled. Systems with more than one input and one output are called multivariable systems.

Idle-Speed Control of an Automobile As another example of a control system, we consider the idle-speed control of an automobile engine. The objective of such a control system is to maintain the engine idle speed at a relatively low value (for fuel economy) regardless of the applied engine loads (e.g., transmission, power steering, air conditioning). Without the idle-speed control, any sudden engine-load application would cause a drop in engine speed that might cause the engine to stall. Thus the main objectives of the idle-speed control system are (1) to eliminate or minimize the speed droop when engine loading is applied and (2) to maintain the engine idle speed at a desired value. Fig. 1-2 shows the block diagram of the idle-speed control system from the standpoint of inputs-system-outputs. In this case, the throttle angle a and the load torque TL (due to the application of air conditioning, power steering, transmission, or power brakes, etc.) are the inputs, and the engine speed to is the output. The engine is the controlled process of the system.

Sun-Tracking Control of Solar Collectors To achieve the goal of developing economically feasible non-fossil-fuel electrical power, the U.S. government has sponsored many organizations in research and development of solar power conversion methods, including the solar-cell conversion techniques. In most of

1-1 Introduction 5

Load torque TL

• Throttle angle a

w

ENGINE Engine speed 0) ^

Figure 1-2 Idle-speed control system.

Figure 1-3 Solar collector field.

these systems, the need for high efficiencies dictates the use of devices for sun tracking. Fig. 1-3 shows a solar collector field. Fig. 1-4 shows a conceptual method of efficient water extraction using solar power. During the hours of daylight, the solar collector would produce electricity to pump water from the underground water table to a reservoir (perhaps on a nearby mountain or hill), and in the early morning hours, the water would be released into the irrigation system.

One of the most important features of the solar collector is that the collector dish must track the sun accurately. Therefore, the movement of the collector dish must be controlled by sophisticated control systems. The block diagram of Fig. 1-5 describes the general philosophy of the sun-tracking system together with some of the most important compo­nents. The controller ensures that the tracking collector is pointed toward the sun in the morning and sends a "start track" command. The controller constantly calculates the sun's rate for the two axes (azimuth and elevation) of control during the day. The controller uses the sun rate and sun sensor information as inputs to generate proper motor commands to slew the collector.

1-1-3 Open-Loop Control Systems (Nonfeedback Systems)

• Open-loop systems are economical but usually inaccurate.

The idle-speed control system illustrated in Fig. 1-2, shown previously, is rather un­sophisticated and is called an open-loop control system. It is not difficult to see that the system as shown would not satisfactorily fulfill critical performance requirements. For instance, if the throttle angle a is set at a certain initial value that corresponds to a certain

SOLAR COLLECTOR

Figure 1-4 Conceptual method of efficient water extraction using solar power.

h Sun's Rate

SUN SENSOR

J i

e

Position Error

CONTROLLER

Torque Disturbance T,i

Trim Rate

•? "

H

- i - >

<

Command Motor Rate

s /, J MOTOR DRIVER

v^DISH COLLECTOR

o \

^ _ ^ L O A D

SPEED REDUCER

«—

n Figure 1-5 Important components of the sun-tracking control system.

engine speed, then when a load torque TL is applied, there is no way to prevent a drop in the engine speed. The only way to make the system work is to have a means of adjusting a in response to a change in the load torque in order to maintain m at the desired level. The conventional electric washing machine is another example of an open-loop control system because, typically, the amount of machine wash time is entirely determined by the judgment and estimation of the human operator.

The elements of an open-loop control system can usually be divided into two parts: the controller and the controlled process, as shown by the block diagram of Fig. 1 -6. An input signal, or command, r, is applied to the controller, whose output acts as the actuating signal u; the actuating signal then controls the controlled process so that the controlled variable y will perform according to some prescribed standards. In simple cases, the controller can be

Reference input /'

CONTROLLER

Actuating signal u CONTROLLED

PROCESS

Controlled variable y

Figure 1-6 Elements of an open-loop control system.

1-1 Introduction < 7

an amplifier, a mechanical linkage, a filter, or other control elements, depending on the nature of the system. In more sophisticated cases, the controller can be a computer such as a microprocessor. Because of the simplicity and economy of open-loop control systems, we find this type of system in many noncritical applications.

1-1-4 Closed-Loop Control Systems (Feedback Control Systems)

What is missing in the open-loop control system for more accurate and more adaptive control is a link or feedback from the output to the input of the system. To obtain more accurate control, the controlled signal y should be fed back and compared with the reference input, and an actuating signal proportional to the difference of the input and the output must be sent through the system to correct the error. A system with one or more feedback paths such as that just described is called a closed-loop system.

• Closed-loop systems have A closed-loop idle-speed control system is shown in Fig. 1-7. The reference input cor

many advantages over open- sets the desired idling speed. The engine speed at idle should agree with the reference value loop systems. cor, and any difference such as the load torque TL is sensed by the speed transducer and the

error detector. The controller will operate on the difference and provide a signal to adjust the throttle angle a to correct the error. Fig. 1-8 compares the typical performances of open-loop and closed-loop idle-speed control systems. In Fig. l-8(a), the idle speed of the open-loop system will drop and settle at a lower value after a load torque is applied. In Fig. 1-8 (b), the idle speed of the closed-loop system is shown to recover quickly to the preset value after the application of TL.

The objective of the idle-speed control system illustrated, also known as a regulator system, is to maintain the system output at a prescribed level.

Error detector

CONTROLLER ENGINE

SPEED TRANSDUCER

Figure 1-7 Block diagram of a closed-loop idle-speed control system.

Desired idle speed

0)r

Application of T,

Time

Desired idle speed

(a)

Application of TL

V

(b)

Time

Figure 1-8 (a) Typical response of the open-loop idle-speed control system, (b) Typical response of the closed-loop idle-speed control system.

8 Chapter 1. Introduction

1-2 WHAT IS FEEDBACK, AND WHAT ARE ITS EFFECTS?

• Feedback exists whenever there is a closed sequence of cause-and-effect relationships.

The motivation for using feedback, as illustrated by the examples in Section 1-1, is somewhat oversimplified. In these examples, feedback is used to reduce the error between the reference input and the system output. However, the significance of the effects of feedback in control systems is more complex than is demonstrated by these simple examples. The reduction of system error is merely one of the many important effects that feedback may have upon a system. We show in the following sections that feedback also has effects on such system performance characteristics as stability, bandwidth, overall gain, impedance, and sensitivity.

To understand the effects of feedback on a control system, it is essential to examine this phenomenon in a broad sense. When feedback is deliberately introduced for the purpose of control, its existence is easily identified. However, there are numerous situations where a physical system that we recognize as an inherently nonfeedback system turns out to have feedback when it is observed in a certain manner. In general, we can state that whenever a closed sequence of cause-and-effect relationships exists among the variables of a system, feedback is said to exist. This viewpoint will inevitably admit feedback in a large number of systems that ordinarily would be identified as nonfeedback systems. However, control-system theory allows numerous systems, with or without physical feedback, to be studied in a systematic way once the existence of feedback in the sense mentioned previously is established.

We shall now investigate the effects of feedback on the various aspects of system performance. Without the necessary mathematical foundation of linear-system theory, at this point we can rely only on simple static-system notation for our discussion. Let us consider the simple feedback system configuration shown in Fig. 1-9, where r is the input signal; y, the output signal; e, the error; and b, the feedback signal. The parameters G and H may be considered as constant gains. By simple algebraic manipulations, it is simple to show that the input-output relation of the system is

M = >- = G

r \+GH 1-1

Using this basic relationship of the feedback system structure, we can uncover some of the significant effects of feedback.

1-2-1 Effect of Feedback on Overall Gain

• Feedback may increase the gain of a system in one frequency range but decrease it in another.

As seen from Eq. (1-1), feedback affects the gain G of a nonfeedback system by a factor of 1 + GH. The system of Fig. 1-9 is said to have negative feedback, because a minus sign is assigned to the feedback signal. The quantity GH may itself include a minus sign, so the general effect of feedback is that it may increase or decrease the gain G. In a practical control system, G and H are functions of frequency, so the magnitude of 1 4- GH may be

i r"i

r

u b

+ -

+ e

-o

- v j

—r"»

- \ J

G

H

o-

o-

o-

o-

C\ 1

y r\ \J

Figure 1-9 Feedback system.

1-2 What Is Feedback, and What Are Its Effects? 9

greater than 1 in one frequency range but less than 1 in another. Therefore, feedback could increase the gain of system in one frequency range but decrease it in another.

1-2-2 Effect of Feedback on Stability

• A system is unstable if its output is out of control.

Stability is a notion that describes whether the system will be able to follow the input command, that is, be useful in general. In a nonrigorous manner, a system is said to be unstable if its output is out of control. To investigate the effect of feedback on stability, we can again refer to the expression in Eq. (1-1). If GH = - 1 , the output of the system is infinite for any finite input, and the system is said to be unstable. Therefore, we may state that feedback can cause a system that is originally stable to become unstable. Certainly, feedback is a double-edged sword; when it is improperly used, it can be harmful. It should be pointed out, however, that we are only dealing with the static case here, and, in general, GH = — 1 is not the only condition for instability. The subject of system stability will be treated formally in Chapters 2, 5, 7, and 8.

It can be demonstrated that one of the advantages of incorporating feedback is that it can stabilize an unstable system. Let us assume that the feedback system in Fig. 1-9 is unstable because GH — — 1. If we introduce another feedback loop through a negative feedback gain of F, as shown in Fig. 1-10, the input-output relation of the overall system is

G y =

r 1 + GH + GF \-T,

• Feedback can improve stability or be harmful to stability.

It is apparent that although the properties of G and H are such that the inner-loop feedback system is unstable, because GH = - 1 , the overall system can be stable by properly selecting the outer-loop feedback gain F. In practice, GH is a function of frequency, and the stability condition of the closed-loop system depends on the magnitude and phase of GH. The bottom line is that feedback can improve stability or be harmful to stability if it is not properly applied.

Sensitivity considerations often are important in the design of control systems. Because all physical elements have properties that change with environment and age, we cannot always consider the parameters of a control system to be completely stationary over the entire operating life of the system. For instance, the winding resistance of an electric motor changes as the temperature of the motor rises during operation. Control systems with electric components may not operate normally when first turned on because

1 Oi -r LJ

r

\ j b

- +

+ e

- +

+ -0

-o

-o

-o

f~\

-\J

G

H

F

O-

O-

O

o-

o-

o-

/-> _1_ U 1

y

t_J —

Figure 1-10 Feedback system with two feedback loops.

10 Chapter 1. Introduction

• Note: Feedback can increase or decrease the sensitivity of a system.

of the still-changing system parameters during warmup. This phenomenon is sometimes called "morning sickness." Most duplicating machines have a warmup period during which time operation is blocked out when first turned on.

In general, a good control system should be very insensitive to parameter variations but sensitive to the input commands. We shall investigate what effect feedback has on sensitivity to parameter variations. Referring to the system in Fig. 1-9, we consider G to be a gain parameter that may vary. The sensitivity of the gain of the overall system M to the variation in G is defined as

dM/M percentage change in M o G -

dG/G percentage change in G 1-3

where dM denotes the incremental change in M due to the incremental change in G, or dG. By using Eq. (1-1), the sensitivity function is written

dM G_ 1 M

cM _

dG 1 +GH '1-4)

This relation shows that if GH is a positive constant, the magnitude of the sensitivity function can be made arbitrarily small by increasing GH, provided that the system remains stable. It is apparent that, in an open-loop system, the gain of the system will respond in a one-to-one fashion to the variation in G (i.e., SQ = 1). Again, in practice, GH is a function of frequency; the magnitude of 1 4- GH may be less than unity over some frequency ranges, so feedback could be harmful to the sensitivity to parameter variations in certain cases. In general, the sensitivity of the system gain of a feedback system to parameter variations depends on where the parameter is located. The reader can derive the sensitivity of the system in Fig. 1 -9 due to the variation of H.

1-2-3 Effect of Feedback on External Disturbance or Noise

• Feedback can reduce the effect of noise.

All physical systems are subject to some types of extraneous signals or noise during operation. Examples of these signals are thermal-noise voltage in electronic circuits and brush or commutator noise in electric motors. External disturbances, such as wind gusts acting on an antenna, are also quite common in control systems. Therefore, control systems should be designed so that they are insensitive to noise and disturbancevs and sensitive to input commands.

The effect of feedback on noise and disturbance depends greatly on where these extraneous signals occur in the system. No general conclusions can be reached, but in many situations, feedback can reduce the effect of noise and disturbance on system performance. Let us refer to the system shown in Fig. 1-11, in which /-denotes the command

+ o-r

- O- b - +

•o o-

o o-

-o o G2

-o o-

•o o H

-o o

-o +

>' - o -

Figure 1-11 Feedback system with a noise signal.

1-3 Types of Feedback Control Systems : 11

signal and n is the noise signal. In the absence of feedback, that is, H= 0, the output)' due to n acting alone is

v = G?n 1-5]

With the presence of feedback, the system output due to n acting alone is

G2 y = 1 + G]G2H

;i-6)

Feedback also can affect Comparing Eq. (1-6) with Eq. (1-5) shows that the noise component in the output of bandwidth, impedance, transient responses, and frequency responses.

Eq. (1-6) is reduced by the factor 1 + GxGiH if the latter is greater than unity and the system is kept stable.

In Chapter 9, the feedforward and forward controller configurations are used along with feedback to reduce the effects of disturbance and noise inputs. In general, feedback also has effects on such performance characteristics as bandwidth, impedance, transient response, and frequency response. These effects will be explained as we continue.

1-3 TYPES OF FEEDBACK CONTROL SYSTEMS

Feedback control systems may be classified in a number of ways, depending upon the purpose of the classification. For instance, according to the method of analysis and design, control systems are classified as linear or nonlinear, and time-varying or time-invariant. According to the types of signal found in the system, reference is often made to continuous-data or discrete-data systems, and modulated or unmodulated systems. Control systems are often classified according to the main purpose of the system. For instance, a position-control system and a velocity-control system control the output variables just as the names imply. In Chapter 9, the type of control system is defined according to the form of the open-loop transfer function. In general, there are many other ways of identifying control systems according to some special features of the system. It is important to know some of the more common ways of classifying control systems before embarking on the analysis and design of these systems.

1-3-1 Linear versus Nonlinear Control Systems

• Most real-life control systems have nonlinear characteristics to some extent.

This classification is made according to the methods of analysis and design. Strictly speaking, linear systems do not exist in practice, because all physical systems are nonlinear to some extent. Linear feedback control systems are idealized models fabricated by the analyst purely for the simplicity of analysis and design. When the magnitudes of signals in a control system are limited to ranges in which system components exhibit linear characteristics (i.e., the principle of superposition applies), the system is essentially linear. But when the magnitudes of signals are extended beyond the range of the linear operation, depending on the severity of the nonlinearity, the system should no longer be considered linear. For instance, amplifiers used in control systems often exhibit a saturation effect when their input signals become large; the magnetic field of a motor usually has saturation properties. Other common nonlinear effects found in control systems are the backlash or dead play between coupled gear members, nonlinear spring characteristics, nonlinear friction force or torque between moving members, and so on. Quite often, nonlinear characteristics are intentionally introduced in a control system to improve its performance

12 Chapter 1. Introduction

or provide more effective control. For instance, to achieve minimum-time control, an on-off (bang-bang or relay) type controller is used in many missile or spacecraft control systems. Typically in these systems, jets are mounted on the sides of the vehicle to provide reaction torque for attitude control. These jets are often controlled in a full-on or full-off fashion, so a fixed amount of air is applied from a given jet for a certain time period to control the attitude of the space vehicle.

• There are no general For linear systems, a wealth of analytical and graphical techniques is available for methods for solving a wide design and analysis purposes. A majority of the material in this text is devoted to the class of nonlinear systems, analysis and design of linear systems. Nonlinear systems, on the other hand, are usually

difficult to treat mathematically, and there are no general methods available for solving a wide class of nonlinear systems. It is practical to first design the controller based on the linear-system model by neglecting the nonlinearities of the system. The designed controller is then applied to the nonlinear system model for evaluation or redesign by computer simulation. The Virtual Lab introduced in Chapter 6 is mainly used to model the characteristics of practical systems with realistic physical components.

1-3-2 Time-Invariant versus Time-Varying Systems

When the parameters of a control system are stationary with respect to time during the operation of the system, the system is called a time-invariant system. In practice, most physical systems contain elements that drift or vary with time. For example, the winding resistance of an electric motor will vary when the motor is first being excited and its temperature is rising. Another example of a time-varying system is a guided-missile control system in which the mass of the missile decreases as the fuel on board is being consumed during flight. Although a time-varying system without nonlinearity is still a linear system, the analysis and design of this class of systems are usually much more complex than that of the linear time-invariant systems.

Continuous-Data Control Systems A continuous-data system is one in which the signals at various parts of the system are all functions of the continuous time variable t. The signals in continuous-data systems may be further classified as ac or dc. Unlike the general definitions of ac and dc signals used in electrical engineering, ac and dc control systems carry special significance in control systems terminology. When one refers to an ac control system, it usually means that the signals in the system are modulated by some form of modulation scheme. A dc control system, on the other hand, simply implies that the signals are unmodulated, but they are still ac signals according to the conventional definition. The schematic diagram of a closed-loop dc control system is shown in Fig. 1-12. Typical waveforms of the signals in response to a step-function input are shown in the figure. Typical components of a dc control system are potentiometers, dc amplifiers, dc motors, dc tachometers, and so on.

Figure 1-13 shows the schematic diagram of a typical ac control system that performs essentially the same task as the dc system in Fig. 1-12. In this case, the signals in the system are modulated; that is, the information is transmitted by an ac carrier signal. Notice that the output controlled variable still behaves similarly to that of the dc system. In this case, the modulated signals are demodulated by the low-pass characteristics of the ac motor. Ac control systems are used extensively in aircraft and missile control systems in which noise and disturbance often create problems. By using modulated ac control systems with carrier frequencies of 400 Hz or higher, the system will be less susceptible to low-frequency noise. Typical components of an ac control system are synchros, ac amplifiers, ac motors, gyroscopes, accelerometers, and so on.

1-3 Types of Feedback Control Systems 13

DC Motor

W c f l J ^ H

0 i 0

Figure 1-12 Schematic diagram of a typical dc closed-loop system.

Q en O AC Synchro transmitter

Synchro control

transformer

W ^ ) e "

o i

Figure 1-13 Schematic diagram of a typical ac closed-loop control system.

In practice, not all control systems are strictly of the ac or dc type. A system may incorporate a mixture of ac and dc components, using modulators and demodulators to match the signals at various points in the system.

Discrete-Data Control Systems Discrete-data control systems differ from the continuous-data systems in that the signals at one or more points of the system are in the form of either a pulse train or a digital code. Usually, discrete-data control systems are subdivided into sampled-data and digital control systems. Sampled-dala control systems refer to a more general class of

14 Chapter 1. Introduction

Figure 1-14 Block diagram of a sampled-data control system.

Digital coded input k DIGITAL

COMPUTER

, .

h,

^

DIGITAL-TO-ANALOG

CONVERTER

ANALOG-TO-DIGITAL

CONVERTER

4

%

i k i w AIRFRAME

C P M C H D C dJDlNdUKo

Attitude of

missile

A

^

w^

Figure 1-15 Digital autopilot system for a guided missile.

• Digital control systems are usually less susceptible to noise.

discrete-data systems in which the signals are in the form of pulse data. A digital control system refers to the use of a digital computer or controller in the system so that the signals are digitally coded, such as in binary code.

In general, a sampled-data system receives data or information only intermittently at specific instants of time. For example, the error signal in a control system can be supplied only in the form of pulses, in which case the control system receives no information about the error signal during the periods between two consecutive pulses. Strictly, a sampled-data system can also be classified as an ac system, because the signal of the system is pulse modulated.

Figure 1-14 illustrates how a typical sampled-data system operates. A continuous-data input signal r{t.) is applied to the system. The error signal e{t) is sampled by a sampling device, the sampler, and the output of the sampler is a sequence of pulses. The sampling rate of the sampler may or may not be uniform. There are many advantages to incorporating sampling into a control system. One important advantage is that expensive equipment used in the system may be time-shared among several control channels. Another advantage is that pulse data are usually less susceptible to noise.

Because digital computers provide many advantages in size and flexibility, computer control has become increasingly popular in recent years. Many airborne systems contain digital controllers that can pack thousands of discrete elements into a space no larger than the size of this book. Figure 1-15 shows the basic elements of a digital autopilot for guided-missile control.

1-4 SUMMARY

In this chapter, we introduced some of the basic concepts of what a control system is and what it is supposed to accomplish. The basic components of a control system were described. By demonstrat­ing the effects of feedback in a rudimentary way, the question of why most control systems are closed-loop systems was also clarified. Most important, it was pointed out that feedback is a double-edged sword—it can benefit as well as harm the system to be controlled. This is part of the challenging task of designing a control system, which involves consideration of such performance criteria as stability,